Welcome to the Institute for Digital Research and Education

Stata Annotated Output
Multinomial Logistic Regression

This page shows an example of an multinomial logistic regression analysis with footnotes explaining the output. The data were collected on 200 high school
students and are scores on various tests, including science, math, reading and social studies. The outcome measure in this analysis is
socio-economic status (ses)- low, medium and high- from which we are going to see what relationships exists with science test scores (science),
social science test scores (socst) and gender (female). Our response variable, ses, is going to be treated as
categorical under the assumption that the levels of ses status have no natural ordering
and we are going to allow Stata to choose the referent group, middle ses. The first half of this page
interprets the coefficients in terms of multinomial log-odds (logits) and the second half interprets the coefficients in terms of
relative risk ratios.

a. This is a listing of the log likelihoods at each iteration. Remember that
multinomial logistic regression, like binary and ordered logistic regression, uses maximum likelihood
estimation, which is an iterative procedure. The first iteration (called iteration 0) is the log
likelihood of the "null" or "empty" model; that is, a model
with no predictors. At the next iteration, the predictor(s) are included in the model. At each iteration, the
log likelihood decreases because the goal is to minimize the log likelihood. When the difference between successive iterations is
very small, the model is said to have "converged", the iterating stops, and the results are displayed.
For more information on this process for binary outcomes, see
Regression Models for Categorical and Limited Dependent Variables by J. Scott Long (page 52-61).

Model Summary

b. Log Likelihood - This is the log likelihood of the fitted model. It is used in the Likelihood Ratio Chi-Square test of whether all predictors'
regression coefficients in the model are simultaneously zero and in tests of nested models.

c. Number of obs - This is the number of observations used in the
multinomial logistic regression.
It may be less than the number of cases in the dataset if there are missing
values for some variables in the equation. By default, Stata does a listwise
deletion of incomplete cases.

d. LR chi2(6) - This is the Likelihood Ratio (LR) Chi-Square test that
for both equations (low ses relative to middle ses and high ses
relative to
middle ses) at
least one of the predictors' regression coefficient is not equal to zero.
The number in the parentheses indicates the degrees of freedom of the Chi-Square distribution
used to test the LR Chi-Square statistic and is defined by the number of models
estimated (2) times the number of predictors in the model (3).
The LR Chi-Square statistic can be calculated by
-2*( L(null model) - L(fitted model)) = -2*((-210.583) - (-194.035)) = 33.096, where L(null model)
is from the log likelihood with just the response variable in the model (Iteration 0)
and L(fitted model) is the log likelihood from the final iteration (assuming the model converged) with all the parameters.

e. Prob > chi2 - This is the probability of getting a LR test statistic as extreme as, or more so, than the observed under the null
hypothesis; the null hypothesis is that all of the regression coefficients
across both models are simultaneously equal to zero. In other words, this is the probability of obtaining this
chi-square statistic (33.10) if there is in fact no effect of the predictor variables. This p-value is compared to a specified alpha level, our willingness
to accept a type I error, which is typically set at 0.05 or 0.01. The small p-value from the LR test, <0.00001, would lead us to conclude that at least
one of the regression coefficients in the model is not equal to zero. The parameter of the Chi-Square distribution used to test the null hypothesis is defined
by the degrees of freedom in the prior line, chi2(6).

f. Pseudo R2 - This is McFadden's pseudo R-squared. Logistic regression does not have an equivalent to the R-squared that is found in OLS regression;
however, many people have tried to come up with one. There are a wide variety of pseudo-R-square statistics. Because this statistic does not mean what
R-square means in OLS regression (the proportion of variance for the response variable explained by the predictors), we suggest interpreting this statistic with great
caution.

g. ses - This is the response variable in the multinomial logistic regression. Underneath ses are two
replicates of the predictor variables, representing the two models that
are estimated: low ses relative to middle ses and high ses
relative to
middle ses.

h and i. Coef. and referent group - These are the estimated
multinomial logistic regression coefficients and the referent level,
respectively, for the model. An important feature of the multinomial logit model
is that it estimates k-1 models, where k is the number of levels
of the dependent variable. In this instance, Stata, by default, set middle ses as the
referent group and therefore estimated a model for low ses relative to middle
ses and a model for high ses relative to middle ses. Therefore, since
the parameter estimates are relative to the referent group, the standard
interpretation of the multinomial logit is that for a unit change in the
predictor variable, the logit of outcome m relative to the referent group
is expected to change by its respective parameter estimate given the variables
in the model are held constant.

low ses relative to middle ses

science - This is the multinomial logit estimate
for a one unit increase in science test score for low ses relative
to middle ses given the other variables in the model are held constant.
If a subject were to increase his science test score by one point, the
multinomial log-odds for low ses relative to middle ses would be
expected to decrease by 0.024 unit while holding all other variables in the
model constant.

socst - This is the multinomial logit estimate
for a one unit increase in socst test score for low ses relative
to middle ses given the other variables in the model are held constant.
If a subject were to increase his socst test score by one point, the
multinomial log-odds for low ses relative to middle ses would be
expected to decrease by 0.039 unit while holding all other variables in the
model constant.

female - This is the multinomial logit estimate
comparing females to males for low ses relative
to middle ses given the other variables in the model are held constant.
The multinomial logit for females relative to males is 0.817 unit higher for
being in low ses relative to middle ses given all other predictor variables in the
model are held constant.

_cons - This is the multinomial logit estimate for
low ses relative to middle ses when the predictor variables in the model
are evaluated at zero. For males (the variable female evaluated at zero)
with zero science and socst test scores, the logit for being in
low ses versus middle ses is 1.912. Note, evaluating science and socst
at zero is out of the range of plausible test scores and if the test scores were
mean-centered, the intercept would have a natural interpretation: log odds of
being in low ses versus middle ses for a male with average science and socst test score.

high ses relative to middle ses

science - This is the multinomial logit estimate
for a one unit increase in science test score for high ses relative
to middle ses given the other variables in the model are held constant.
If a subject were to increase his science test score by one point, the
multinomial log-odds for high ses relative to middle ses would be
expected to increase by 0.023 unit while holding all other variables in the
model constant.

socst - This is the multinomial logit estimate
for a one unit increase in socst test score for high ses relative
to middle ses given the other variables in the model are held constant.
If a subject were to increase his socst test score by one point, the
multinomial log-odds for high ses relative to middle ses would be
expected to increase by 0.043 unit while holding all other variables in the
model constant.

female - This is the multinomial logit estimate
comparing females to males for high ses relative
to middle ses given the other variables in the model are held constant.
The multinomial logit for females relative to males is 0.033 unit lower for
being in high ses relative to middle ses given all other predictor variables in the
model are held constant.

_cons - This is the multinomial logit estimate for
high ses relative to middle ses when the predictor variables in the model
are evaluated at zero. For males (the variable female evaluated at
zero) with zero science and socst test scores, the logit for being in
high ses relative to middle ses is -4.057.

j. Std. Err. - These are the standard errors of the individual
regression coefficients for the two respective models estimated. They are used
in both the calculation of the z test
statistic, superscript k, and the confidence interval of the regression coefficient, superscript
l.

k. z and P>|z| - These are the test statistics and p-value, respectively,
that within a given model the
null hypothesis that an individual predictor's regression
coefficient is zero given that the rest of the predictors are in the model. The test statistic z is the ratio of the Coef. to the
Std. Err. of the respective predictor. The z value follows a standard normal distribution which is used to test against a two-sided
alternative hypothesis that the Coef. is not equal to zero. The probability that a particular z test statistic is as extreme as, or more
so, than what has been observed under the null hypothesis is defined by P>|z|.
The interpretation of the parameter estimates' significance is limited only to the
first equation, low ses relative to middle ses. The interpretation
for the second model, high ses relative to middle ses, naturally falls out of the first
equations interpretation.

For low ses relative to middle ses, the z test statistic for the predictor science (-0.024/0.021) is
-1.12 with an associated p-value of 0.261. If we set our
alpha level to 0.05, we would fail to reject the null hypothesis and conclude that
for low ses relative to middle ses, the regression coefficient for science
has not been found to be statistically different from zero given socst and female are in the model.
For low ses relative to middle ses, the z test statistic for the predictor socst (-0.039/0.020) is
-1.99 with an associated p-value
of 0.046. If we again set our alpha level to 0.05, we would reject the null hypothesis and conclude that the regression coefficient for socst has
been found to be statistically different from zero for low ses relative
to middle ses given
that science and female are in the model.
For low ses relative to middle ses, the z test statistic for the predictor
female (0.817/0.391) is 2.09 with an associated p-value
of 0.037. If we again set our alpha level to 0.05, we would reject the null hypothesis and conclude that the
difference between males and females has been found to be statistically
different for low ses relative to middle ses given
that science and female are in the model.
For low ses relative to middle ses, the z test statistic for the
intercept, _cons (1.912/1.129) is 1.70 with an associated p-value
of 0.090. With an alpha level of 0.05, we would fail to reject the
null hypothesis and conclude, a) that the multinomial logit for males (the
variable
female evaluated at zero) and with zero science and socst
test scores in low ses relative to middle ses are found not to be
statistically different from zero; or b) for males with zero science and
socst test scores, you are statistically uncertain whether they are more
likely to be classified as low ses or middle ses. We can make the
second interpretation when we view the _cons as a specific covariate
profile (males with zero science and socst test scores). Based on the direction and significance of
the coefficient, the _cons tells whether the profile would have a greater
propensity to fall in one of the levels of the dependent variable.

l. [95% Conf. Interval] - This is the Confidence Interval (CI) for an individual
multinomial logit regression coefficient given the other predictors are in the model
for outcome m relative to the referent group.
For a given predictor with a level of 95% confidence, we'd say that we are 95% confident that the "true" population
multinomial logit regression coefficient lies
between the lower and upper limit of the interval for outcome m
relative to the referent group. It is calculated as the Coef. ± (zα/2)*(Std.Err.), where zα/2
is a critical value on the standard normal distribution. The CI is equivalent to the z test statistic: if the CI includes zero, we'd fail to
reject the null hypothesis that a particular regression coefficient is zero given the other predictors are in the model.
An advantage of a CI is that it is illustrative; it provides a range where the "true" parameter may lie.

Relative Risk Ratio Interpretation

The following is the interpretation of the multinomial logistic regression in terms of
relative risk ratios and can be obtained by
mlogit, rrr after running the multinomial logit model or by specifying the rrr option
when the full model is specified. This part of the interpretation applies to the output below.

a. Relative Risk Ratio - These are the relative risk ratios for the
multinomial logit model shown earlier. They can be obtained by exponentiating
the multinomial logit coefficients, ecoef., or by specifying the rrr option.
Recall that the multinomial logit model estimates k-1 models, where the kth equation is relative to the referent group. If the model
was to be written out in an exponentiated form where the predictor of interest
is evaluated at x + δ and at x for outcome m relative to
referent group, where
δ is the change in the predictor we are interested
in (δ is traditionally is set to one) while the other variables in the
model are held constant. If we then take their ratio, the ratio would reduce to the ratio
of two probabilities, the relative risk. In this sense, the exponentiated
multinomial logit coefficient provides an estimate of relative risk. However,
the exponentiated coefficient are commonly interpreted as odds
ratios. Standard interpretation of
the relative risk ratios is for a unit change in the predictor variable, the
relative risk ratio of
outcome m relative to the referent group is expected to change by a
factor of the respective parameter estimate given the variables in
the model are held constant.

low ses relative to middle ses

science - This is the relative risk ratio for a one unit
increase in science score for low ses relative to middle ses
level given that the other variables in the model are held constant. If a
subject were to increase her science test score by one unit, the
relative risk for low ses relative to middle ses would be expected to
decrease by a factor
of 0.977 given the other variables in the model are held constant. So, given a
one unit increase in science, the relative risk of being in the low
ses group would be 0.977 times more likely when the other variables in the
model are held constant. More generally, we can say that if a subject were to
increase their science test score, they'd be
expected to fall into middle ses as compared to low ses.

socst - This is the relative risk ratio for a one
unit increase in socst score for low ses relative to middle ses
level given that the other variables in the model are held constant. If a
subject were to increase her socst test score by one unit, the
relative risk for low ses relative to middle ses would be expected to decrease by a factor
of 0.962 given the other variables in the model are held constant.

female - This is the relative risk ratio comparing
females to males for low ses relative to middle ses
level given that the other variables in the model are held constant. For females
relative to males, the relative risk for low ses relative to middle ses would be expected to increase by a factor
of 2.263 given the other variables in the model are held constant.

high ses relative to middle ses

science - This is the relative risk ratio for a one unit
increase in science score for high ses relative to middle ses
level given that the other variables in the model are held constant. If a
subject were to increase her science test score by one unit, the
relative risk for high ses relative to middle ses would be expected to increase by a factor
of 1.023 given the other variables in the model are held constant.

socst - This is the relative risk ratio for a one
unit increase in socst score for high ses relative to middle ses
level given that the other variables in the model are held constant. If a
subject were to increase their socst test score by one unit, the
relative risk for high ses relative to middle ses would be expected to increase by a factor
of 1.043 given the other variables in the model are held constant.

female - This is the relative risk ratio comparing
females to males for high ses relative to middle ses
level given that the other variables in the model are held constant. For females
relative to males, the relative risk for high ses relative to middle ses would be expected to
decrease by a factor
of 0.968 given the other variables in the model are held constant.

b. [95% Conf. Interval] - This is the CI for the relative risk ratio
given the other predictors are in the model. For a given predictor with a level
of 95% confidence, we'd say that we are 95% confident that the "true" population
relative risk ratio comparing outcome m to the referent group lies
between the lower and upper limit of the interval. An advantage of a CI is that it is
illustrative; it provides a range where the "true" relative risk ratio may lie.